Real-rooted atlas

A038319 → x ↦ x+1 → D

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(32 + 10\,x + 16\,n + 5\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 160 + 50\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
016050210
1384024003756615
28192076800240002500185220
316384002048000960000200000156254862025
43145728049152000307200009600000150000093750122523030
5587202560110100480086016000035840000084000000105000005468753001814235
610737418240234881024002202009600011468800000358400000067200000070000000312500072043541640
7193273528320483183820800528482304000330301440000129024000000322560000005040000000450000000175781251702028671245

All 15 rows

Technical details
Node id
n-a224d395cadb1de0 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-833b492313a93e6e

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            160,              50;
  1:           3840,            2400,             375;
  2:          81920,           76800,           24000,            2500;
  3:        1638400,         2048000,          960000,          200000,           15625;
  4:       31457280,        49152000,        30720000,         9600000,         1500000,          93750;
  5:      587202560,      1101004800,       860160000,       358400000,        84000000,       10500000,         546875;
  6:    10737418240,     23488102400,     22020096000,     11468800000,      3584000000,      672000000,       70000000,       3125000;
  7:   193273528320,    483183820800,    528482304000,    330301440000,    129024000000,    32256000000,     5040000000,     450000000,     17578125;
  8:  3435973836800,   9663676416000,  12079595520000,   8808038400000,   4128768000000,  1290240000000,   268800000000,   36000000000,   2812500000,    97656250;
  9: 60473139527680, 188978561024000, 265751101440000, 221459251200000, 121110528000000, 45416448000000, 11827200000000, 2112000000000, 247500000000, 17187500000, 537109375;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (16*n+32+(5*n+10)*x)*P(n-1,x) for n >= 1, with P(0,x) = 160 + 50*x.
%t Mathematica
P[0, x_] := 160 + 50*x;
P[n_, x_] := P[n, x] = Expand[Together[((16*n + 32 + (5*n + 10)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 160 + 50*x; for(n = 1, N, my(i = n + 1); v[i] = ((16*n + 32 + (5*n + 10)*x)*v[i - 1])/(n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.